Vietnam Journal of Mechamcs, VAST, Vol.35, No 1 (2013), pp 5 1 - 6 5
A CO-ROTATIONAL B E A M E L E M E N T F O R G E O M E T R I C A L L Y N O N L I N E A R
ANALYSIS OF P L A N E F R A M E S
N g u y e n Dinh K i e n \ Trinh T h a n h H u o n g S Le Thi Ha^
'^Institute of Mechamcs, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam
^Hanoi University of Transport and Communication, Vietnam
A b s t r a c t . A co-rotational beam element for geometrically nonlinear analysis of plane frames is presented. On the base of the shallow arch expression for local strain, the ele- ment IS formulated by using exact polynomials to interpolate the transversal displacement and rotation. Using the formulated element, the cntical load and equilibrium path ai'e computed with the aid of the bracketing proceduie and the arc-length control method, respectively. Numerical examples show that the pioposed element is capable to give ac- cuiate results with a smaller number of elements comparing to the elements previously used in the examples. The effect of the nonlinear term used in the local strain expression on the numerical lesults is also investigated and highlighted
Keywords. Plane frame, geometric nonlinearity, t o-rotational beam element, critical load, equilibrium path,
1. I N T R O D U C T I O N
Frame structure is widely used in civil engineering, and nonlinear analysis of this structure is of interest from both academic and practical points of views. The essential for a nonlinear analysis is to have a nonlinear element which is capable to model accurately nonlinear behavior of the structure. Development of nonlinear elements, in general, and beam elements, in particular, is an important topic in the field of structural mechanics.
Many beam elements which included geometrical nonlinearity for the large displacement analysis of frames can be found in the literature, and some of them have been documented in various well-known textbooks, e.g. [1.2] Depending on the choice of reference configu- ration, the nonlinear beam elements can be classified into three main types, namely the total Lagrangian formulation, the updated Lagrangian formulation and the co-rotational formulation.
In the co-rotational formulation, which will be adopted in the present work, a local coordinate system attached to the element is introduced, and the finite element formulation is firstly formulated in the local coordinates and then transferred to the global system by the aid of transformation matrices [2,3]. Literature survey shows that the co-rotational beam elements can be formulated by using different local strain definitions, beam theories
52 Nguyen Dmh Kien, Tnnh Thanh Huong, Le Tht Ha
as well as the way to define a local coordinate system. In addition to the plane Bernoulh element using a linear strain definition described by Crisfield in [2], the formulations proposed by Pacoste and Eriksson [3], Hsiao and co-workers [4], Meek and Xue [5], are some main co-rotational elements available in the literature for geometrically nonlmear analysis of plane elastic beam and frame structures. In a recent work, Jafari and co- workers [6] also adopted an element attached coordinate system for construction of the flexibility matrix for geometrically nonlinear Timoshenko beam element, which can be viewed as a form of the co-rotational formulation. It is necessary to note that the total Lagrangian formulation and the co-rotational formulation, as demonstrated in [3], though adopted different reference systems for the kinematic description, they are identical.
In the previous work, the first author and his co-worker have conducted some re- search on large displacement analysis of elastic and elasto-plastic frames by developing the co-rotational Bernoulli beam elements [7,8]. The effect of shear deformation on the large displacement response of frames has also investigated by the first author by formulation of the co-rotational Timoshenko element [9]. In the context of the finite element analy- sis, due to the independence of the transversal displacement and rotation, a Timoshenko beam element for geometrically nonlinear analysis of frames, as discussed by Pacoste and Eriksson in [3] or by the first author in [9.10], can be formulated by linearly interpolat- ing all the kinematic variables, provided that some techniques such as the reduced-order integration are employed to prevent the formulated element from the shear locking. An alternative way for preventing a shear deformable beam element from the shear locking problem, as discussed by Reddy in [11], is to employ appropriate interpolation functions for the kinematic variables. The exact cubic and quadratic polynomials obtained by solv- ing the homogenous equilibrium equations of a beam element proposed by Luo in [12], which will be adopted in the present work, are examples of such interpolation functions.
The present paper formulates a co-rotational element for geometrically nonlinear analysis of plane frame structure. The element is adopted a shallow arch expression for local strain and employing exact polynomials obtained from the field consistent approach to interpolate the transversal displacement and rotation. Using the formulated element, nonlinear equilibrium equations in the finite element context are constructed and solved by an incremental and iterative method procedure. The bracketing procedure and the arc- length control method are adopted in computing the critical loads and tracing equihbriura paths, respectively. A number of numerical examples are given to demonstrate the accuracy and efficiency of the formulated element. The effect of the nonlinear term in the local strain expression on the accuracy of numerical results is also investigated and highlighted.
2. C O - R O T A T I O N A L M E T H O D
As above mentioned, in the co-rotational method an element attached coordinate system which continuously translates and rotates with the element during its deformation process is introduced. By using such a local system, the element deformation is decom- posed into rigid body and pure deformation parts, and the geometric nonlinearity induced by large rigid body motion is incorporated into transformation matrices which can be derived from the relationship between the local and global quantities [2,3]. The kinematic
A co-rotalional beam element for geometrically nonlinear analysis of plane frames 53
relationship and detail expressions for the transformation matrices are described in this section.
defcimed conriguralion
Fig. 1. A plane beam element and its kinematics in local and global coordinate systems
Fig. 1 shows a two-node plane beam element and its kinematics in the local and global coordinate systems. The element is assumed initially straight and the beam material IS linearly elastic. Global coordinates {X. Z) are fixed in space. A local system (x. z) is attached to the element, and it continuously translates and rotates with the element.
Denoting (X,, Zi) and {Xj. Z^) are respectively the coordinates of node i and node j with respective to the global system. The local system [x, z) is chosen as well as its original IS always at node i, and x axis directs to node j . By choosing such a local system, the axial displacement at node i and transversal displacements at both the nodes in the local system are always zero. Thus, vectors of the nodal displacements in tbe local and global systems are given by, respectively,
di = {ut 9u ^ o T (!) and
d = {t(, w, 9i Uj Wj Oj]^ (2) Here and afterwards a superscript T denotes the transpose of a vector or a matrix, and
subscript "/" stands for the "'locaV.
The local displacements in Eq. (1) can be computed as [2,3]
Ui=ln-l- 9u^e,-9r. 9ij =9j-9r (3) where /. /„ and 6^ are the initial length, current length and rigid rotation of the element,
respectively. These quantities can be computed by geometrical consideration as l=y/ll-+ll. ln = Vilx + d4l)^ + Hz + ds2V
lxdb2 ~ Iz^Al
With Ix =Xj-X„lz = Zj- Z, are the projections of undeformed element on the global axes X and Z, respectively: du ~ % — Un dz2 = lOj - ^ i .
a^ Nguyen Dmh Kien, Tnnh Thanh Huong, Le Thi Ha
Denoting fm, and f,n are the element internal force vectors corresponding to the nodal displacement vectors defmed by Eq. (1) and Eq. (2), respectively. These force vectors contain the axial, shear forces and moments at the nodes
f.„, ^{AT, Mu Mi,f (5)
^n. = {Ni Q, M, Nj Qj Mjf (6) Assuming the strain energy expression for the element U has been derived in terms of the
nodal displacements, the global internal force vector can be computed by differentiation of this strain energy (which is invariant of the coordinate systems) with respect to the global displacement vector as
dU aUdd, _ rSU Tf, IJ,
^ " " S d - f l d y s d ^ ^ ' a d , ' • ^ ' ' ' " ' ^' where Ti is the transformation matrix, which can be computed from Eq. (3) as
The element tangent stiffness matrix in the global system kt can be computed as
kt = ^ -^ ^ = T r k t , T i + N{T2 + MUT:^ + MijT, (9) where kt, = d'^U/ddf is the local tangent stiffness matrix; T2, T3 and T4 are the trans-
formation matrices, defined from Eq (3) as
T3 = T , = T . = - ^ (11)
with Or is defined by Eq. (4). Thus, Eqs. (7)-(ll) completely define the element formulation provided that; the local internal force vector f,„, and tangent stiffness matrix kt, are known.
3. L O C A L F O R C E V E C T O R A N D S T I F F N E S S M A T R I X This section derives the local internal force vector and tangent stiffness matrix for the element. Adopting the Timoshenko beam theory, the displacements in the local coordinate system are given by
U{X.Z) = UQ{X) ~ z9{x) , w{x.z)-w{x) . v{x,z) = Q (12) where uo{x) and UJ(X) are the mid-plane axial and transversal displacements; 9{x) is the cross sectional rotation, independent ofthe transversal displacement w{x). The shallow arch expression, a degenerated form of the Green strain, deduced from Eq. (12), is adopted for the local strain as
dua ^ 1 faw\^ 39 au, From Eq. (3), one gets
A co-Tolational beam element for geometrically nonlmear anaiysts of plane frames 55
where \ = —d9/dx is the beam curvature. The strain energy for the beam element asso- ciated with the strain components defined by Eq. (13) has the form
U = l- [ EAtl^dx + \ j Elx^dx +\ [ KGA-t^dx (14)
^ Jo ^ Jo ^ Jo
where EA. EI and GA arc the axial, bending and shear rigidities, respectively; K is the shear correction factor, depends on the cross section geometry.
In order to derive the local internal force vector and tangent stiffness matrix from the strain energy expression, we need to express the train energy in term of the local nodal displacements. To this end, interpolation functions are necessary to introduce for the axial and transverse displacements and the cross section rotation. In the present work.
the field consistence approach discussed by Luo [12] is adopted, in which the interpolation functions for the transversal displacement w(x) and the rotation 9{x) are found by solving the homogeneous equilibrium equations of the beam element
^""^[-a-x + ^J
a^e ( Sw\ (15) with the boundary conditions
mU=o = "'I. . S|x=o = Sl, . w\i=i = «i|j . 9\i=i = e,j (16) By introducing a dimensionless parameter
^ 12 EI
and noticing that wu = wi, = 0, Eqs. (15) and (16) give the solutions w = Nu,iei, + .V„29|j
e = NnOi, + Ni,2«l,
{17}
(18) where
;
' 1+4
'•^nr^iil)
" 2 \l) 2 l\( j ) ' - ( 4 + p ) f + ( ! + « 1
(19)
(20)
Tho axial displacement is still linearly interpolated as uo = j U | . and thus to prevent the formulated element from the membrane locking which might be occurred from the
Nguyen D,nh K,en, TVinA Thanh Huong, Le Th, Ha
unbalance interpolation, the membrane strain £„ in Eq. (13) is necessary to replace by the effective strain as [2]
1 r J I f' auoi (aw\
dx (21)Using Eqs. (19) and (20) we can write the effective strain in terms of the local nodal displacements as
1 , 1 \<t>{2 + 4'),
-ui- ksi. - 0„f + i(29?, - »,.«,, + e?,)] (22)
'"' ^ ; " ' " ( ! + # [ 24
Substituting Eqs. (19), (20) mto Eq. (14), and using Eq. (22) one can express the strain energy in term of the local nodal displacements as
, , IBA(1 ^ 1 \<i>{2 + 4>),
C ^ —-— < T " ( + •; koii - e,,f + ^(2«i'i - SuOl, + 29?i)| }
+
2 \ ( ' (1 + ?.)H 24
'^' • [*(2 + 4>)(eu - e„f + i(el + e^e,, +»?,)] + ^ ^ ^ ( » , . + e,,f (23)
2({1 + 4>y ^'^^ ' ^ " " '" ' " " " ''" 8 ( 1 + cj,)^'
Expressions for the local internal force vector fi„, and tangent stiffness matrix kt, are then obtained by successive differentiation of the strain energy expression, Eq. (23), with respective to the local degrees of freedom as
f,„ au
ad, and k,,
a^u
(24)From Eqs. (23) and (24), one can write the components of the local internal force vector, fjn,. in the explicit forms as
Nl = EAe,B
IE A [ 0 ( 2 + j.) 1
+ 7 ( l f ^ l*(2 + m , -»,) + 2(2.,. + e,)] + ^ | ^ ( « „ + e,) p^, Ml, lEA f(»(2 + <f.) 1
12 ( * 0 - « « ) + ^ ( 4 « , i - % ) | £ e f r . (1 + *
+ 7 ( I T ^ l'''(2 + *)(«0 - «'.) + 2(9,. + 201,)] + ^f^{eu + «y) where Ces is given by Eq. (22). The local tangent stiffness matrix, kt,, can be written in the form
kt, = k , + kh + k, (26) in which
0 0 0 • 0 {^>'^ + 2<t, + i) -((1,^+24,-2) 0 - ( 1 * 2 ^ 2 , ^ - 2 ) (<t>^+2^1 + 4) kb = EI
i{i+'t>y
IIJI'K.GA
^4(l + ,^)2 0 0 0
0 1 1
0' 1 1
(27)
wtationai beam element JOT geometrically nonl,near analysis of plane {ram
are the constant stiffness matrices stemming from the beam bending and the shear de- formation, respectively: ka is the nonlinear stiffness matrix resulted from the membrane strain, and having the form
"1 1 (5^.^ + 10(^ + 8
/
60(1 + 4,f
1 (50^ + lOQi.-F2)
60(1-^.^)2 ^' (502 + lQ(f, + 8)
60(1 + 0)2 '^' (28)
t, Ofeff.
-mr = ^ H T T T : ^ E^^(2 -K m i - Bij) + 2(4^a - ^ , ) ] d9u 6 0 ( 1 + 0 ) : 1 [5<t>{2 + (l>){9ij-9u) + 2i49tj-9u)]
(29) - d9ij 6 0 ( 1 + 0 ) 2 ^
The element formulated from the strain energy defined by Eq. (23) is based on the nonlinear shallow arch expression for the local axial strain, Eq. (13). However, a linear expression for the local strain can be assumed for the local strain [2,3], and the element based on the linear strain definition can be obtained by deleting the terms in the Eq. (23) corresponding to the nonlinear term ~{dw/dx)^. In order to investigate the effect of the nonhnear term on the numerical results, we denote B2CL and B2CS as the elements based on the linear and shallow arch expressions of the local strain, respectively.
It is necessary to note that when 0 —> 0, the strain energy given by Eq (23) deduces to the strain energy expression of the shallow arch based co-rotational Bernoulli beam element, previously derived by Pacoste and Eriksson in [3] (conf Eq. (44) of Ref [3]). Thus, with the definition of the coordinate systems as that in [3]. when 0 —• 0 the Timoshenko beam element of the present work deduces exactly to the 2D co-rotational Bernoulli element in [3]. In this regards, the element of the present work is able to model the slender beams and frames. In other work, the Timoshenko beam element formulated in present paper is free of the shear locking.
4. N U M E R I C A L P R O C E D U R E S
The equilibrium equations for non-linear analysis of structures are obtained by set- ting the out-of-balance forces to zeros (2]
g ( p . A ) - q m ( p ) - A f e f - 0 (30) whore the out-of balance force vector g is a function of the current structural nodal dis-
placements p. and the load-level parameter A: q,n is the structural internal force vector.
constructed from the element force vector f,, in standard way of the finite element method.
f,-f is the fixed external loading vector. The non-linear equations (30) can be solved by an inrrnnental/iterative technique based on the Newton-Raphson method, in which the norm of vector g(p. A) is guided towards zero.
58 Nguyen Dmh Kien, TVinh Thanh Huong, Le Thi Ha
In order to trace complete equilibrium paths for possible complex behavior such as snap-through and snap-back, the arc-length method developed by Crisfield [2,13] is employed. In the method, the load parameter A becomes a variable, and a constraint equation is introduced as
a - ( A p ^ A p + AA2v.2f,^fef) - A / ' = 0 (31) where A/ is a fixed value chosen by the analyst: the vector A p and scalar AA are the
incremental displacement and load parameter, defined from the last converged state; ^ is the scaling parameter. With ;/J = 1 , Eq. (31) defines a spherical surface with a radius A/ in the load-displacement space, and Eq. (31) define a cylindrical surface when t/i = 0.
The methods corresponding to t/^ = 1 and V = 0 are known in the literature as the
"spherical arc-length method'" and "'cylindrical arc-length method", respectively. Thus, with the introduction of the constrain equation, instead of solving the equifibrium equations (30) directly, we find the intersection of the equilibrium path with the surface defined by Eq. (31). The iterative procedure is obtained from the truncated Taylor expansion of the vector g and a around the current equilibrium point as
g = gjo + %oSp + I f |o5A ^ g|o + Kt|o<5p + U{6X = 0
d p dX (32) a = ao + 2Ap^i5p -I- 2AA(5Ai/'^feTfef = 0
where g|o and a[o are defined from the current equilibrium point. Eq. (32) gives the iterative displacements i5p and iterative load parameter 5\ as
As we see the augmented Jacobian matrix within the square brackets in Eq. (33) remains non-singular even when the Kt is singular. However, the Jacobian matrix in (33) is neither symmetric nor banded, and the computation based on the arc-length method is expensive.
The cylindrical arc-length method is adopted in the present work. More detail on the arc-length method and its numerical implementation are described in [2].
In addition, the bracketing procedure [14,15] is adopted in computing critical and limit loads of the structures. To this end, a load-control strategy is employed in solving the equilibrium equation (30), and the lowest eigenvalue r of the structural tangent stiffness matrix Kt. assembled from the element matrix kt, is employed as indicator for predicting a critical or a limit point When passing the lowest critical or hmit point, predicted by the negative sign of r, the bracketing procedure is active to compute an intermediate point Aj, by using an interpolation scheme for the control parameter A as
Ai = A , - ^ ^ - ^ T , (34) Tr -Ti
where A;, A^ are the values ofthe load-level parameter at the equilibrium points before and after passing the critical or limit point, respectively; r/ and r-r are the lowest eigenvalues of Kt at the points. The bracketing procedure is terminated when a convergency criteria
A ca-Totational beam element for geometrically Tionlmear analysis of plane frames 59
is satisfied. The following criterion is adopted herewith
where 0 is the tolerance factor, taken by 10"'* in the present work. The obtained value A^
from the bracketing procedure is taken as the critical load-level parameter.
5. N U M E R I C A L E X A M P L E S
In this section, various beam and frame structures shown in Fig. 2 are analyzed to verify the formulated element. A tolerance factor of 10"^ is also chosen for judging convergency of the iterative numerical procedure described in section 4. The shear modulus is taken as G = E/2 for all the problem, and if not mentioned otherwise, the shear correction factor K — 5/6 is chosen for all the analyses reported below.
5.1. Euler buckling load
The cantilever beam under a centric axial force P with geometric and material data shown in Fig 2(a) adopted from Ref [3] is employed to study the convergency and accuracy ofthe formulated element in computing critical loads. The buckling coefficient ofthe beam, fn = PcL'^/EI with Pc is the critical load, computed by various numbers of the B2CL and B2CS elements is listed in Tab. 1. The analytical solution [16], Pc = Pe/{l+Pe/nGA) with Pg = •K^jAEIjL'^, gives the buckling coefficient / B = 2.4674. As seen from Tab. 1, the accuracy of the critical load obtained by the B2CS can be accepted at a vcr>' coarse mesh. The accuracy of the critical load computed by the B2CL element is steady improved by using more elements in the analysis, but the load computed by twenty B2CL elements just gives the same accuracy as that obtained by two B2CS elements. Thus, it is hardly improved the accuracy of the critical load by using a large number of elements when the nonlinear term is ignored from the local strain expression.
,. I
Fig. 2. Test problems for verifying formulated element
Following the work by Pacoste and Eriksson in [3], the critical load ofthe cantilever beam is computed by different slenderness ratios, namely s = 4 and s = 100. where
60 Nguyen Dmh Kien, Tnnh Thanh Huong, Le Tht Ha
s = Lj-JTjA. Keeping all the data as in Fig. 2(a), the computation is performed with the beam cross-sectional area A = 2.88 x 10"^ and ^ = 0.018, respectively.
Table 1. Buckhng coefficient for cantilever beam computed by different element numbers
Number of element 1 2 4 6 8 10 20
Element B2CL 3.0000 2.5966 2.4993 2.4815 2.4753 2.4725 2.4687
B2CS 2.4860 2.4687 2.4675 2.4674 2.4674 2.4674 2.4674
Table 2. Critical load for cantilever beam at, different slenderness ratios
s 4 100
Pc
lie
Pc
Uc
Present B2CL 550.7782
19.1242 444.9728 0.0247
B2CS 548.1241
19.0321 444.0597 0.0247
Ref. [3]
b2dtt 553.0 14.18 446.8 0.0248
b2dcs 548.2 19.03 444.6 0.02477
Ref. [17|
548.64 444.13
In consistent with the work in [3], the beam is discretized by 8 elements. The critical load and the corresponding axial displacement at free end of the beam obtained by the B2CL and B2CS elements are listed in Tab. 2. For comparison, the critical loads obtained from a numerical method [3], and from a finite strain theory [17] are also listed in the ' table. In the table, the •b2dtt" and •b2dcs' are the Timoshenko and Bernoulli beam el- ements, formulated in [3] by using the total Lagrangian and co-rotational formulations, respectively. The numerical result of the present work is very good in agreement with the result computed by the •b2dcs' of Ref [3], and with that of [17]. As has been shown in [3], the •b2dcs' is verj' accurate in computing the critical load and large displacement response of planar beam and frame structures. It is noted that for the case s = 4, a shear correction factor -0 = 1200 as adopted in [3] is chosen in the computation to inhibit the shear effect.
5.2. Cantilever b e a m u n d e r t r a n s v e r s a l p o i n t load
The cantilever beam subjected to a transversal point load as shown in Fig. 2(b) is employed herewith to test the ability of the elements in tracing the equilibrium path for large displacement. The problem has been analytically analyzed by Mattiasson [18], and it has been employed by several authors to test the accuracy of nonlinear beam elements.
A co-rotational beam element for geometrically notilmear analysis of plane frames 61
The load-displacement curves computed by only two elements formulated in the present work are shown in Fig. 3, where the analytical solution derived by Mattiasson [18] and the finite element result computed by five consistent Bernoulli beam elements by Nanakom and Vu [19] are also depicted. As seen from the figure, the numerical result computed by
Fig. 3. Load-displacement relation cantilever beam under transversal point load
only two B2CS elements of the present work agrees very well with the analytical solution in [18], The curves computed by the B2CL element slightly differs from that obtained by the B2CS element. The difference in the numerical result computed by the B2CL and B2CS elements in this example can be eliminated by using 4 or more elements 5.3, Cantilever beam under eccentric axial point load
The cantilever beam with the data shown in Fig 2(c) under an eccentric axial point load is considered herewith. The problem has been previously analyzed by Wood and Zienkiewicz by using five paralinear elements [20]. The load-displacement curves obtained by using a mesh of three elements formulated in the present work are depicted in Fig 4.
^
s;a J,
"8
1 ^
o o
7^
o WooO a ZianklewBZ 3B2CL8lBn,Bms 3 B2CS elemenls
\ .
>',
-
Fig. 4. Load-displacement relation for cantilever beam under eccentric axial load
6S Nguyen Dtnh Kien, Tnnh Thanh Huong, Le Thi Ha
The good agreement between the result computed by the B2CS element of the present work with the numerical result obtained in [20] is observed. Some difference between the curves computed by the B2CL element with that obtained by the B2CS element is seen in the figure. This difference can be diminished by using more elements in the analysis, and with a mesh of 5 elements, no difference is observed.
5.4. W i l l i a m s toggle frame
The toggle frame in Fig. 2(d) was previously employed by several authors to test the behavior of nonlinear finite elements in large displacement analysis of frames [3,4,19]. The frame data are: L = 12.943. H = 0.386. b = 0.753. h = 0.243. E = 10.3 x 10^. The previous work shows snap-through behavior of the frame under the downwards point load P. The load-deflection curves for the frame computed by various numbers ofthe B2CL and B2CS elements are depicted in Fig. 5. The numerical result obtained by using ten paralinear
Fig. 5. Load-displacement curve for Williams frame
Table 3. Limit load around snap-through situation of Williams toggle frame
Number of element 2 4 6 8 10 12
Element B2CL 39.9904 39.9253 35.8876 34.4337 33.9811 33.6169
B2CS 33.0079 32.9903 32 8587 32.8409 32.8270 32.8066
elements of Wood and Zienkiewicz [20] is also shown in the figure by small triangles. Very good agrement between the curve computed by two B2CS element of the present work and
A co-rotattonai beam element for geometncally noniinear analysis of plane frames 63
that in [20] is noted. The two B2CL elements gives the curve, which largely differs from the corresponding curve obtained by the two B2CS and the numerical result of Ref [20].
To improve the accuracy, a larger number of the B2CL is required, and as seen from Fig.
5 the curve computed by eight B2CL elements has been improved considerately.
To examine the effect of the nonlinear term ^ ( - ^ 1 ^ , the limit load around the snap-through situation of the frame is computed with different numbers of the the B2CL and B2CS elements by using the bracketing procedure. The obtained result listed Tab. 3 shows that the limit load is overestimated by using the B2CL element, even with a fine mesh of twelve B2CL elements.
5.5. Lee's frame
An asymmetric frame under a concentrated load shown in Fig, 2(e). which also known in the literature as the Lee's frame, is investigated. The problem has been previously analyzed by Pacoste and Eriksson [3], Hsiao and Huo [4], and Cichon [21]. The frame shows highly nonlinear with snap-through and snap-back behavior. The frame data are L = 120 cm, A = 6 cm2, I == 2 cm'*, E = 720 t/cm^, where L, A, I and E are the length, cross section area, moment of inertia and elastic modulus, respectively. A model of five elements with different length as shown in Fig. 2(e) is employed in the present work. The displacements in both horizontal and vertical directions at the loaded point are computed by using the B2CS element.
Fig. 6. Load-displacement relation of Lee's frame
Fig. 6 shows the relation between the applied load and the displacements at loaded point ofthe frame. The computed curves ofthe present work agrees well with the numerical result obtained by using 10 elements in Ref [4], which shown in the figure by small circles.
As in case of the Williams toggle frame, the curves computed by the B2CL (not shown in the figure) differ from the curves in Fig 6. To improve the numerical result, more elements are required when the B2CL element is used.
In order to show the ability in modelling the shear deformation, the computation was performed with various values of the slenderness ratio, namely s = 150.50. and 30 by
Nguyen Dmh Kien, Tnnh Thanh Huong, Le Thi Ha
Fig. 7. Load-displacement relation of Lee's frame with different slenderness ratios
using five B2CS elements. Keeping the rest data as above, the computation is performed with A = 3.125. 0.3472 and 0.1250 cm'^, respectively. The computed curves depicted in Fig. 7 show a reduction in the limit load which the frame can withstand when reducing the slenderness ratio. Thus, the shear deformation effect on the large displacement behavior of the frame can be well modelled by the formulated element.
6. C O N C L U S I O N S
In this paper, a co-rotational beam element for geometrically nonlinear analysis of plane frames has been formulated. The shallow arch expression was adopted for the local strain and the exact polynomials obtained from the field consistence approach were employed as interpolation functions for the transversal displacement and sectional rotation.
Using the formulated element, the critical loads and equilibrium paths are computed with the aid of the bracketing procedure and the arc-length method. A number of examples have been given to demonstrate the accuracy and efficiency of the formulated element.
The numerical results have shown that the proposed element is capable to give accurate results by using a smaller number of the elements comparing to the elements previously used in the examples.
The effect of nonlinear term in the local strain expression on the critical loads and large displacement behavior of the structure has also been numerically investigated. It has been shown that a larger number of elements is required to ensure the accuracy when the nonlinear term in the expression of the local strain is ignored. In addition, ignoring the nonlinear term leads to an overestimation of the critical and limit loads, and in some cases this shortcoming is hardly improved by increasing number of the elements used in analvsis.
A C K N O W L E D G E M E N T
The financial support from Vietnam NAFOSTED to the present work is gratefully acknowledged.
A co-rotational beam element for geometncally nonlinear analysis of plane frames 65
REFERENCES
[I] T. Belytschko, W . K . Liu a n d B. Moran, Nonlinear fimte elements for continua and structures (John Wiley &; Sons, Chichester, 2000).
[2] M A. Crisfield, Non-linear finite element analysis of solids and structures, Vol. 1: Essentials (John Wiley & Sons, Chichester, 1991)
[3] C. Pacoste and A. Eriksson, Beam elements in instability problems, Comput Method. Appl Mech. Eng., 144, (1997), 163-197
[4] K M Hsiao and F . Y Huo, Nonhnear finite element analysis of elastic frames, Comput. Struct., 26, (1987), 693-701.
[5] J L. Meek a n d Q. Xue, A study on the instability problem for 2D-frames, Comput. Method Appl Mech. Eng., 1 3 6 (1996), 347-361.
[6] V Jafari, S.H. Vahdani a n d M. Rahimian, Derivation of the consistent flexibility mtrix for geometrically nonhnear Timoshenko frame fmite element, Finite Elem. Anal. Des., 46, (2010), 1077-1085.
[7] N.D.Kien and D Q Quang, Large deflections analysis of frames by elements contaning higer- order terms, Vietnam Joumal of Mechanics, 2 5 , (2003), 243-254.
|8] N.D.Kien and D.Q. Quang, Beam element for large displacement analysis of elasto-plastic frames, Vietnam Joumal of Mechanics, 2 6 , (2004), 39-54,
[9] N.D.Kien, Effect of shear defromation on large deflection behavior of elastic frames, Vietnam Journal of Mechanics, 2 6 , (2004) 167-181.
[10] D,K. Nguyen, Post-buckling behavior of beam on two-parameter elastic foundation. Int. J.
Struct. Stabil. Dynam., 4, (2004), 21-43.
[II] .I-N, Reddy, On locking-free shear deformable beam fmite elements, Comput. Method Appl.
Mech , Eng., 1 4 9 , (1997), 113-132.
[12] Y H. Luo, Explanation and elimination of shear locking and membrane locking with field consistence approach, Comput. Method. Appl. Mech Eng., 162, (1998), 249-269.
[13] M A. Crisfield, A fast incremental/iterative solution procedure t h a t handles 'snap-through', CompuL Struct., 1 3 , (1981), 55-62.
[14] M.A. Crisfield, Non-linear finite element analysis of solids and structures. Volume 2: Advanced topics, John Wiley & Sons, Chichester, (1997)
[15] J. Shi, Computing critical points and secondary paths in nonlinear structural stability analysis by the finite element method, Comput. Struct, 5 8 , (1996), 203-220
[16] S.P. Timoshenko and J.M. Cere, Theory of elastic stability (McGraw-Hill, New York, 1961).
[17] Y Goto, T . Yoshimitsu and M. Obara, Elliptic integrals solution of plane elastica with axial and shear deformations, Int J. Solids Struct., 26, (1990), 375-390.
[18] K. Mattiasson, Numerical results for large deflection beam and frame problems analysed by means of elliptic integrals, Int J. Numer. Methods Eng., 17, (1981), 145-153
[19] P. Nanakorn and L.N. Vu, A 2D field-consistent beam element for large displacement analysis using the total Lagrangian formulation, Finite Elem. Anal. Des., 4 2 , (2006), 1240-1247 [20] R.D. Wood and O.C. Zienkiewicz, Geometrically nonhnear finite element analysis of beams,
frames, arches a n d axisymmetric shells, Comput. Struct., 7, (1977), 725-735.
[21] C. Cich6n, Large displacement in-plane analysis of elastic-plastic frames, Comput. Struct., 19, (1984), 737-745.
Received November 14, 2011